Renewal structure and local time for diffusions in random environment

TL;DR

This paper analyzes the local time behavior of a one-dimensional diffusion in a drifted Brownian potential, characterizing the limit laws of the supremum of local time and favorite sites using stable Lévy processes.

Contribution

It introduces an extended renewal structure to explicitly describe the asymptotic distribution of local times and favorite sites in the diffusion.

Findings

Limit law of the supremum of local time characterized.

Position of favorite sites described explicitly.

Asymptotic behavior linked to a two-dimensional stable Lévy process.

Abstract

We study a one-dimensional diffusion $X$ in a drifted Brownian potential $W\_\kappa$, with $ 0\textless{}\kappa\textless{}1$, and focus on the behavior of the local times $(\mathcal{L}(t,x),x)$ of $X$ before time $t\textgreater{}0$.In particular we characterize the limit law of the supremum of the local time, as well as the position of the favorite sites. These limits can be written explicitly from a two dimensional stable L{\'e}vy process. Our analysis is based on the study of an extension of the renewal structure which is deeply involved in the asymptotic behavior of $X$.